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diff --git a/Secondo anno/Elementi di probabilitÃ  e statistica/sections/3-probabilitÃ -reale.tex b/Secondo anno/Elementi di probabilitÃ  e statistica/sections/3-probabilitÃ -reale.tex
index 93311c0..25a52b4 100644
--- a/Secondo anno/Elementi di probabilitÃ  e statistica/sections/3-probabilitÃ -reale.tex	
+++ b/Secondo anno/Elementi di probabilitÃ  e statistica/sections/3-probabilitÃ -reale.tex	
@@ -683,6 +683,7 @@ che $\mu$ assuma valori finiti per le funzioni semplici).
 \end{lemma}
 
 \begin{theorem}[di convergenza monotona, o di Beppo Levi]
+    \label{th:convergenza_monotona}
     Sia $(X_i)_{i \in \NN}$ una successione di v.a.~reali non negative q.c.~con
     $X_i \goesup X$ q.c.~(cioÃ¨ la successione Ã¨ crescente e
     $X_i(\omega) \to X(\omega)$ $P$-quasi ovunque). Allora $\EE[X_i] \goesup \EE[X]$. \smallskip
@@ -701,4 +702,19 @@ che $\mu$ assuma valori finiti per le funzioni semplici).
     Segue la stessa idea di dimostrazione per il teorema nella sua forma per l'integrale di Lebesgue.
 \end{theorem}
 
+\subsection{Calcolo del valore atteso}
+
+\begin{proposition}
+    Sia $X : \Omega \to \RR$ una v.a.~assolutamente continua con densitÃ  $f$
+    e sia $\varphi : \RR \to \RR$ una funzione
+    boreliana. Allora valgono le seguenti affermazioni:
+    \begin{enumerate}[(i.)]
+        \item $\varphi(X)$ Ã¨ integrabile se e solo se $\int_\RR \abs{\varphi(x)} f(x) \dx$ Ã¨ finito.
+        \item se $\varphi(X)$ ammette valore atteso, allora $\EE[\varphi(X)] = \int_\RR \varphi(x) f(x) \dx$.
+    \end{enumerate}
+    Il risultato segue considerando in ordine a) le funzioni indicatrici, b) le funzioni semplici,
+    c) le funzioni non negative e d) le funzioni integrabili, applicando il
+    \hyperref[th:convergenza_monotona]{Teorema di convergenza monotona}. 
+\end{proposition}
+
 \end{multicols*}
\ No newline at end of file
diff --git a/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-modelli-discreti.tex b/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-modelli-discreti.tex
index 4877056..526ad73 100644
--- a/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-modelli-discreti.tex	
+++ b/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-modelli-discreti.tex	
@@ -4,21 +4,22 @@
 \chapter*{Tabella e proprietÃ  delle distribuzioni discrete}
 \addcontentsline{toc}{chapter}{Tabella e proprietÃ  delle distribuzioni discrete}
 
-\vskip -0.3in
+\vskip -0.45in
 
 \begin{table}[htb]
 \label{tab:distr_discrete}
-\scalebox{0.74}{
-\begin{tabular}{|l|l|l|l|l|l|l|}
-\hline
-Nome distribuzione                                                                                             & Caso di utilizzo                                                                                                                                                                                                                                           & Parametri                                                                                                                                                     & DensitÃ  discreta                                                                                                                                 & Valore atteso                                                                         & Momento secondo                   & Varianza                                                                                                   \\ \hline
-\begin{tabular}[c]{@{}l@{}}Distr. di Bernoulli\\ $X \sim B(\pp)$\end{tabular}                                  & \begin{tabular}[c]{@{}l@{}}Esperimento con esito\\ di successo ($1$) o\\ insuccesso ($0$).\end{tabular}                                                                                                                                                    & $\pp$ -- probabilitÃ  di successo.                                                                                                                             & $P(X=1) = \pp$, $P(X=0) = 1-\pp$                                                                                                                     & $\EE[X] = \pp$                                                                          & $\EE[X^2] = \pp$                    & $\Var(X) = \pp(1-\pp)$                                                                                         \\ \hline
-\begin{tabular}[c]{@{}l@{}}Distr. binomiale\\ $X \sim B(n, \pp)$\end{tabular}                                  & \begin{tabular}[c]{@{}l@{}}In una serie di $n$ esperimenti\\ col modello delle prove ripetute,\\ $X$ conta il numero di successi.\\ $X$ Ã¨ in particolare somma di $n$\\ v.a.~i.i.d.~distribuite come $B(\pp)$.\end{tabular}                      & \begin{tabular}[c]{@{}l@{}}$n$ -- numero di esperimenti\\ $\pp$ -- probabilitÃ  di successo\\ dell'$i$-esimo esperimento\end{tabular}                          & \begin{tabular}[c]{@{}l@{}}$P(X=k) = \binom{n}{k}{\pp}^k (1-\pp)^{n-k}$\\ per $0 \leq k \leq n$ e $0$ altrimenti.\end{tabular}                         & \begin{tabular}[c]{@{}l@{}}$\EE[X] = n \pp$\\ (Ã¨ somma di $n$ Bernoulliane)\end{tabular} & $\EE[X^2] = n \pp + n(n-1)\pp^2$       & \begin{tabular}[c]{@{}l@{}}$\Var(X) = n \pp(1-\pp)$\\ (Ã¨ somma di $n$\\ Bernoulliane indipendenti)\end{tabular} \\ \hline
-\begin{tabular}[c]{@{}l@{}}Distr. binomiale\\ negativa\\ $X \sim \BinNeg(h, \pp)$\end{tabular}                 & \begin{tabular}[c]{@{}l@{}}In una serie di infiniti esperimenti\\ col modello delle prove ripetute,\\ $X$ conta l'esperimento in cui si\\ ha l'$h$-esimo successo.\end{tabular}                                                                            & \begin{tabular}[c]{@{}l@{}}$h$ -- numero dei successi da misurare\\ $\pp \in (0, 1)$ -- probabilitÃ  di successo\\ dell'$i$-esimo esperimento\end{tabular}                & \begin{tabular}[c]{@{}l@{}}$P(X=k) = \binom{k-1}{h-1} \pp^h (1-\pp)^{k-h}$\\ laddove definibile e $0$ altrimenti.\end{tabular}                   & \begin{tabular}[c]{@{}l@{}}$\EE[X] = \frac{h}{\pp}$\\ (Ã¨ somma di $h$ Geometriche)\end{tabular}                                                                & $\EE[X^2] = \frac{h(1+h-\pp)}{\pp^2}$ & \begin{tabular}[c]{@{}l@{}}$\Var(X) = \frac{h(1-\pp)}{\pp^2}$ \\ (Ã¨ somma di $h$\\Geometriche indipendenti)\end{tabular}                                                                             \\ \hline
-\begin{tabular}[c]{@{}l@{}}Distr. geometrica\\ $X \sim \Geom(\pp)$\end{tabular}                                & \begin{tabular}[c]{@{}l@{}}In una serie di infiniti esperimenti\\ col modello delle prove ripetute,\\ $X$ conta l'esperimento in cui si\\ ha il primo successo. Ãˆ pari a\\ $\BinNeg(1, \pp)$\end{tabular}                                                  & \begin{tabular}[c]{@{}l@{}}$\pp \in (0, 1)$ -- probabilitÃ  di successo\\ all'$i$-esimo esperimento.\end{tabular}                                                         & \begin{tabular}[c]{@{}l@{}}$P(X=k) = \pp (1-\pp)^{k-1}$ per\\ $k \geq 1$ e $0$ per $k=0$.\end{tabular}                                               & $\EE[X] = \frac{1}{\pp}$                                                                & $\EE[X^2] = \frac{2-\pp}{\pp^2}$      & $\Var(X) = \frac{1-\pp}{\pp^2}$                                                                                \\ \hline
-\begin{tabular}[c]{@{}l@{}}Distr. ipergeometrica\\ $X \sim H(N, N_1, n)$\end{tabular}                          & \begin{tabular}[c]{@{}l@{}}In un'estrazione di $n$ palline in\\ un'urna di $N$ palline, di cui\\ $N_1$ sono rosse, $X$ conta\\ il numero di palline rosse estratte.\end{tabular}                                                                           & \begin{tabular}[c]{@{}l@{}}$N$ -- numero di palline nell'urna\\ $N_1$ -- numero di palline rosse\\ nell'urna\\ $n$ -- numero di palline estratte\end{tabular} & \begin{tabular}[c]{@{}l@{}}$P(X=k) = \frac{\binom{N_1}{k} \binom{N-N_1}{n-k}}{\binom{N}{n}}$\\ laddove definibile e $0$ altrimenti.\end{tabular} &                                                                                       &                                   &                                                                                                            \\ \hline
-\begin{tabular}[c]{@{}l@{}}Distri.di Poisson\\ (o degli eventi rari)\\ $X \sim \Poisson(\lambda)$\end{tabular} & \begin{tabular}[c]{@{}l@{}}In una sequenza di $n \gg 1$\\ esperimenti di parametro $\pp \ll 1$\\ con $n \pp \approx \lambda$,\\ $X$ misura il numero di successi.\\ Si puÃ² studiare come distribuzione\\ limite della distribuzione binomiale.\end{tabular} & $\lambda$ -- tasso di successo.                                                                                                                               & $P(X=k) = \frac{\lambda^k}{k!} e^{-\lambda}$                                                                                                           & $\EE[X] = \lambda$                                                                    & $\EE[X^2] = \lambda(\lambda + 1)$             & $\Var(X) = \lambda$                                                                                        \\ \hline
-\end{tabular}}
+\scalebox{0.85}{
+    \begin{tabular}{|l|l|l|l|l|l|}
+        \hline
+        Nome distribuzione                                                                                             & Caso di utilizzo                                                                                                                                                                                                                                            & Parametri                                                                                                                                                     & DensitÃ  discreta                                                                                                                                 & Valore atteso e momenti                                                                                                                   & Varianza                                                                                                                 \\ \hline
+        \begin{tabular}[c]{@{}l@{}}Distr. di Bernoulli\\ $X \sim B(\pp)$\end{tabular}                                  & \begin{tabular}[c]{@{}l@{}}Esperimento con esito\\ di successo ($1$) o\\ insuccesso ($0$).\end{tabular}                                                                                                                                                     & $\pp$ -- probabilitÃ  di successo.                                                                                                                             & $P(X=1) = \pp$, $P(X=0) = 1-\pp$                                                                                                                 & $\EE[X] = \EE[X^2] = \pp$                                                                                                                 & $\Var(X) = \pp(1-\pp)$                                                                                                   \\ \hline
+        \begin{tabular}[c]{@{}l@{}}Distr. binomiale\\ $X \sim B(n, \pp)$\end{tabular}                                  & \begin{tabular}[c]{@{}l@{}}In una serie di $n$ esperimenti\\ col modello delle prove ripetute,\\ $X$ conta il numero di successi.\\ $X$ Ã¨ in particolare somma di $n$\\ v.a.~i.i.d.~distribuite come $B(\pp)$.\end{tabular}                                 & \begin{tabular}[c]{@{}l@{}}$n$ -- numero di esperimenti\\ $\pp$ -- probabilitÃ  di successo\\ dell'$i$-esimo esperimento\end{tabular}                          & \begin{tabular}[c]{@{}l@{}}$P(X=k) = \binom{n}{k}{\pp}^k (1-\pp)^{n-k}$\\ per $0 \leq k \leq n$ e $0$ altrimenti.\end{tabular}                   & \begin{tabular}[c]{@{}l@{}}$\EE[X] = n \pp$\\ (Ã¨ somma di $n$ Bernoulliane) \\ $\EE[X^2] = n \pp + n(n-1)\pp^2$ \end{tabular}             & \begin{tabular}[c]{@{}l@{}}$\Var(X) = n \pp(1-\pp)$\\ (Ã¨ somma di $n$\\ Bernoulliane indipendenti)\end{tabular}          \\ \hline
+        \begin{tabular}[c]{@{}l@{}}Distr. binomiale\\ negativa\\ $X \sim \BinNeg(h, \pp)$\end{tabular}                 & \begin{tabular}[c]{@{}l@{}}In una serie di infiniti esperimenti\\ col modello delle prove ripetute,\\ $X$ conta l'esperimento in cui si\\ ha l'$h$-esimo successo.\end{tabular}                                                                             & \begin{tabular}[c]{@{}l@{}}$h$ -- numero dei successi da misurare\\ $\pp \in (0, 1)$ -- probabilitÃ  di successo\\ dell'$i$-esimo esperimento\end{tabular}     & \begin{tabular}[c]{@{}l@{}}$P(X=k) = \binom{k-1}{h-1} \pp^h (1-\pp)^{k-h}$\\ laddove definibile e $0$ altrimenti.\end{tabular}                   & \begin{tabular}[c]{@{}l@{}}$\EE[X] = \frac{h}{\pp}$\\ (Ã¨ somma di $h$ Geometriche) \\ $\EE[X^2] = \frac{h(1+h-\pp)}{\pp^2}$ \end{tabular} & \begin{tabular}[c]{@{}l@{}}$\Var(X) = \frac{h(1-\pp)}{\pp^2}$ \\ (Ã¨ somma di $h$\\Geometriche indipendenti)\end{tabular} \\ \hline
+        \begin{tabular}[c]{@{}l@{}}Distr. geometrica\\ $X \sim \Geom(\pp)$\end{tabular}                                & \begin{tabular}[c]{@{}l@{}}In una serie di infiniti esperimenti\\ col modello delle prove ripetute,\\ $X$ conta l'esperimento in cui si\\ ha il primo successo. Ãˆ pari a\\ $\BinNeg(1, \pp)$\end{tabular}                                                   & \begin{tabular}[c]{@{}l@{}}$\pp \in (0, 1)$ -- probabilitÃ  di successo\\ all'$i$-esimo esperimento.\end{tabular}                                              & \begin{tabular}[c]{@{}l@{}}$P(X=k) = \pp (1-\pp)^{k-1}$ per\\ $k \geq 1$ e $0$ per $k=0$.\end{tabular}                                           & \begin{tabular}[c]{@{}l@{}}$\EE[X] = \frac{1}{\pp}$\\ $\EE[X^2] = \frac{2-\pp}{\pp^2}$\end{tabular}                                       & $\Var(X) = \frac{1-\pp}{\pp^2}$                                                                                          \\ \hline
+        \begin{tabular}[c]{@{}l@{}}Distr. ipergeometrica\\ $X \sim H(N, N_1, n)$\end{tabular}                          & \begin{tabular}[c]{@{}l@{}}In un'estrazione di $n$ palline in\\ un'urna di $N$ palline, di cui\\ $N_1$ sono rosse, $X$ conta\\ il numero di palline rosse estratte.\end{tabular}                                                                            & \begin{tabular}[c]{@{}l@{}}$N$ -- numero di palline nell'urna\\ $N_1$ -- numero di palline rosse\\ nell'urna\\ $n$ -- numero di palline estratte\end{tabular} & \begin{tabular}[c]{@{}l@{}}$P(X=k) = \frac{\binom{N_1}{k} \binom{N-N_1}{n-k}}{\binom{N}{n}}$\\ laddove definibile e $0$ altrimenti.\end{tabular} &                                                                                                                                           &                                                                                                                          \\ \hline
+        \begin{tabular}[c]{@{}l@{}}Distri.di Poisson\\ (o degli eventi rari)\\ $X \sim \Poisson(\lambda)$\end{tabular} & \begin{tabular}[c]{@{}l@{}}In una sequenza di $n \gg 1$\\ esperimenti di parametro $\pp \ll 1$\\ con $n \pp \approx \lambda$,\\ $X$ misura il numero di successi.\\ Si puÃ² studiare come distribuzione\\ limite della distribuzione binomiale.\end{tabular} & $\lambda$ -- tasso di successo.                                                                                                                               & $P(X=k) = \frac{\lambda^k}{k!} e^{-\lambda}$                                                                                                     & \begin{tabular}[c]{@{}l@{}}$\EE[X] = \lambda$\\ $\EE[X^2] = \lambda(\lambda + 1)$\end{tabular}                                            & $\Var(X) = \lambda$                                                                                                      \\ \hline
+    \end{tabular}
+}
 \end{table}
 
 Valgono inoltre le seguenti altre proprietÃ :
diff --git a/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-modelli-reali.tex b/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-modelli-reali.tex
index 2fd7d9f..04583ed 100644
--- a/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-modelli-reali.tex	
+++ b/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-modelli-reali.tex	
@@ -4,31 +4,31 @@
     \chapter*{Tabella e proprietÃ  delle distribuzioni assolutamente continue}
     \addcontentsline{toc}{chapter}{Tabella e proprietÃ  delle distribuzioni assolutamente continue}
     
-    \vskip -0.3in
+    \vskip -0.45in
 
     \begin{center}
         \begin{table}[htb]
-            \scalebox{1.1}{
-                \begin{tabular}{|l|l|l|l|l|l|}
-                \hline
-                Nome distribuzione                                                                 & Caso di utilizzo                                                                                         & Parametri                                                                              & DensitÃ                                                                         & Funzione di ripartizione                                                                                                                                                                                                                          & ProbabilitÃ                              \\ \hline
-                \begin{tabular}[c]{@{}l@{}}Distr. uniforme\\ $X \sim U(B)$\end{tabular}            & \begin{tabular}[c]{@{}l@{}}Estrazione di un\\ punto reale a caso\\ su $B$ senza preferenze.\end{tabular} & \begin{tabular}[c]{@{}l@{}}$B \in \BB(\RR)$ -- insieme\\ da cui estrarre.\end{tabular} & $f(x) = \frac{1}{m(B)} 1_B(x)$                                                 & $F(x) = \frac{m((-\infty, x] \cap B)}{m(B)}$                                                                                                                                                                                    & $P(X \in A) = \frac{m(A \cap B)}{m(B)}$ \\ \hline
-                \begin{tabular}[c]{@{}l@{}}Distr. esponenz.\\ $X \sim \Exp(\lambda)$\end{tabular}  & \begin{tabular}[c]{@{}l@{}}Processo di Poisson\\ in senso continuo.\end{tabular}                         & \begin{tabular}[c]{@{}l@{}}$\lambda > 0$ -- parametro\\ di Poisson.\end{tabular}       & $f(x) = \lambda e^{-\lambda x} 1_{(0, \infty)} (x)$                            & \begin{tabular}[c]{@{}l@{}}$F(x) = 1-e^{-\lambda x}$\\ per $x \geq 0$, $0$ altrimenti.\end{tabular}                                                                                                                             &                                         \\ \hline
-                \begin{tabular}[c]{@{}l@{}}Distr. gamma\\ $X \sim \Gamma(r, \lambda)$\end{tabular} & \begin{tabular}[c]{@{}l@{}}Estensione della distr.\\ binomiale in senso\\ continuo.\end{tabular}         & $r > 0$, $\lambda > 0$.                                                                & $f(x) = \frac{\lambda^r}{\Gamma(r)} x^{r-1} e^{-\lambda x} 1_{(0, \infty)}(x)$ &                                                                                                                                                                                                                                 &                                         \\ \hline
-                \begin{tabular}[c]{@{}l@{}}Distr. normale\\ $X \sim N(m, \sigma^2)$\end{tabular} &                                                                                                          & \begin{tabular}[c]{@{}l@{}}$m$ -- media.\\ $\sigma^2 > 0$ -- varianza.\end{tabular}  & $f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\nicefrac{(x-m)^2}{2\sigma^2}}$   & \begin{tabular}[c]{@{}l@{}}$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\nicefrac{t^2}{2}} \dt$\\ per $N(0, 1)$ e si standardizza\\ le altre distr. con il cambio\\ di var. $z = \nicefrac{(x-m)}{\sigma}$.\end{tabular} &                                         \\ \hline
+            \scalebox{1}{
+                \begin{tabular}{|l|l|l|l|l|l|l|}
+                    \hline
+                    Nome distribuzione                                                                 & Caso di utilizzo                                                                                         & Parametri                                                                              & DensitÃ                                                                         & Funzione di ripartizione                                                                                                                                                                                                          & \begin{tabular}[c]{@{}l@{}}Valore atteso \\ e momenti\end{tabular}                                                                                                                                                     & Varianza                                                                                \\ \hline
+                    \begin{tabular}[c]{@{}l@{}}Distr. uniforme\\ $X \sim U(B)$\end{tabular}            & \begin{tabular}[c]{@{}l@{}}Estrazione di un\\ punto reale a caso\\ su $B$ senza preferenze.\end{tabular} & \begin{tabular}[c]{@{}l@{}}$B \in \BB(\RR)$ -- insieme\\ da cui estrarre.\end{tabular} & $f(x) = \frac{1}{m(B)} 1_B(x)$                                                 & $F(x) = \frac{m((-\infty, x] \cap B)}{m(B)}$                                                                                                                                                                                      & \begin{tabular}[c]{@{}l@{}}$\EE[X] = \frac{a+b}2$ se\\ $B = (a, b)$\end{tabular}                                                                                                                                       & \begin{tabular}[c]{@{}l@{}}$\Var(X) = \frac{(b-a)^2}{12}$\\ se $B = (a,b)$\end{tabular} \\ \hline
+                    \begin{tabular}[c]{@{}l@{}}Distr. esponenz.\\ $X \sim \Exp(\lambda)$\end{tabular}  & \begin{tabular}[c]{@{}l@{}}Processo di Poisson\\ in senso continuo.\end{tabular}                         & \begin{tabular}[c]{@{}l@{}}$\lambda > 0$ -- parametro\\ di Poisson.\end{tabular}       & $f(x) = \lambda e^{-\lambda x} 1_{(0, \infty)} (x)$                            & \begin{tabular}[c]{@{}l@{}}$F(x) = 1-e^{-\lambda x}$\\ per $x \geq 0$, $0$ altrimenti.\end{tabular}                                                                                                                               & $\EE[X] = \frac{1}{\lambda}$                                                                                                                                                                                           & $\Var(X) = \frac{1}{\lambda^2}$                                                         \\ \hline
+                    \begin{tabular}[c]{@{}l@{}}Distr. gamma\\ $X \sim \Gamma(r, \lambda)$\end{tabular} & \begin{tabular}[c]{@{}l@{}}Estensione della distr.\\ binomiale in senso\\ continuo.\end{tabular}         & $r > 0$, $\lambda > 0$.                                                                & $f(x) = \frac{\lambda^r}{\Gamma(r)} x^{r-1} e^{-\lambda x} 1_{(0, \infty)}(x)$ &                                                                                                                                                                                                                                   & $\EE[X] = \frac{r}{\lambda}$                                                                                                                                                                                           & $\Var(X) = \frac{r}{\lambda^2}$                                                         \\ \hline
+                    \begin{tabular}[c]{@{}l@{}}Distr. normale\\ $X \sim N(m, \sigma^2)$\end{tabular}   &                                                                                                          & \begin{tabular}[c]{@{}l@{}}$m$ -- media.\\ $\sigma^2 > 0$ -- varianza.\end{tabular}    & $f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\nicefrac{(x-m)^2}{2\sigma^2}}$     & \begin{tabular}[c]{@{}l@{}}$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\nicefrac{t^2}{2}} \dt$\\ per $N(0, 1)$ e si standardizza\\ le altre distr. con il cambio\\ di var. $z = \nicefrac{(x-m)}{\sigma}$.\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\EE[X] = m$. Inoltre\\ $\EE[(X-m)^p] = 0$ se\\ $p$ Ã¨ dispari\\ e $\EE[(X-m)^p] =$ \\ $= \sigma^p(p-1)!!$\\ se $p$ Ã¨ pari, dove\\ $!!$ indica il\\ \textit{doppio fattoriale}.\end{tabular} & $\Var(X) = \sigma^2$                                                                    \\ \hline
                 \end{tabular}
             }
-            \end{table}
+        \end{table}
     \end{center}
 
     Si ricorda che la funzione $\Gamma$ Ã¨ definita in modo tale che $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \dt$; si tratta
-    di un'estensione della nozione di fattoriale ai valori reali (infatti, $\Gamma(n+1) = n!$ per $n \in \NN$).
+    di un'estensione della nozione di fattoriale ai valori reali (infatti, $\Gamma(n+1) = n!$ per $n \in \NN$). \smallskip
     
-    % Valgono inoltre le seguenti altre proprietÃ :
-    %
-    % \small
-    % \begin{itemize}
-    %     \item
-    % \end{itemize}
+    Valgono inoltre le seguenti altre proprietÃ :
+    
+    \small
+    \begin{itemize}
+        \item La distribuzione $\Exp(\lambda)$ coincide con la distribuzione $\Gamma(1, \lambda)$.
+    \end{itemize}
     
     \end{landscape}
\ No newline at end of file
diff --git a/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-phi.tex b/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-phi.tex
index 441fcf3..9ff131d 100644
--- a/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-phi.tex	
+++ b/Secondo anno/Elementi di probabilitÃ  e statistica/sections/tabella-phi.tex	
@@ -1,6 +1,6 @@
 %--------------------------------------------------------------------
-\chapter*{Tabella e proprietÃ  della f.d.r.~\texorpdfstring{$\Phi(x)$}{Î¦(x)} della normale standard \texorpdfstring{$N(0, 1)$}{N(0, 1)}}
-\addcontentsline{toc}{chapter}{Tabella e proprietÃ  della f.d.r.~\texorpdfstring{$\Phi(x)$}{Î¦(x)} della normale standard \texorpdfstring{$N(0, 1)$}{N(0, 1)}}
+\chapter*{Tabella e proprietÃ  della f.d.r.~\texorpdfstring{$\Phi(x)$}{Î¦(x)} della distr.~normale standard \texorpdfstring{$N(0, 1)$}{N(0, 1)}}
+\addcontentsline{toc}{chapter}{Tabella e proprietÃ  della f.d.r.~\texorpdfstring{$\Phi(x)$}{Î¦(x)} della distr.~normale standard \texorpdfstring{$N(0, 1)$}{N(0, 1)}}
 
 Per $a \in \RR$ si definisce la funzione $\Phi(a) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^a e^{-\nicefrac{x^2}{2}} \dx$. L'integrale $\Phi(\infty) \defeq \frac{1}{\sqrt{2\pi}} \int_{-\infty}^a e^{-\nicefrac{x^2}{2}} \dx$ vale
 esattamente $1$, mentre $\Phi(-\infty) \defeq 0$. Per la paritÃ  di $e^{-\nicefrac{x^2}{2}}$ vale che $\Phi(a) = 1 - \Phi(-a)$ (\textbf{simmetria}). A partire da questa funzione si puÃ² calcolare